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If both product and sum of 2 positive reals are integers, are these reals necessarily integers individually?

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Asish Mahapatra ·2009-12-27 08:44:58

(2+âˆš3)(2-âˆš3) = 1 = integer

2+âˆš3 + 2-âˆš3 = 4 = integer

but (2+âˆš3), (2-âˆš3) are not integers

simplest argument of contradiction

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abcd ·2009-12-31 06:41:29

i completely agree with bonne annee

this question doesnt make any sense because you will have infinite contradictory examples

Reason:

there exist infinite no of irrational nos between two rational nos
and the conjugate pair of any irrational no will satify both your conditions
i.e there will be infinite number of solutions which contradict the special condition of BOTH the nos having to be NECESSARILY INTEGERS

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Basic Integral Calculus Operators Symbols Matrix Cases

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω

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